Glashow-Salam-Weinberg Electroweak Model

Scope of tutorial

In this tutorial, we consider the 1st generation of particles, the fermion-vector boson interactions, the fermion-Higgs boson interactions, the Higgs boson self-interactions, and the vector boson interactions. The quantum chromodynamics (QCD) Lagrangian, which describes the interactions between quarks and gluons, is briefly mentioned at the end.

Introduction

In 1934, Enrico Fermi proposed the following 4-fermion model

GF2(ψ¯eγμψν¯)(ψ¯pγμψn),
to describe β-decay processes such as np+e+ν¯e. The quantity ψ¯ψγ0 is called the Dirac conjugate of the spinor ψ, where the operator is the Hermitian conjugate, i.e., the transpose and complex conjugate and γμ are 4×4 matrices defined by the anti-commutation relations γμγν+γνγμ=2gμν, where gμν is the metric tensor of special relativity. GF is called the Fermi constant.

Fermi assumed that β-decay involved vector currents, ψ¯γμψ, but in 1956 T. D. Lee and C. N. Yang noted that no experiment had been performed to test the assumption of parity invariance, that is, the invariance of particle interactions with respect to a reflection in a mirror. They suggested that certain experimental puzzles could be explained if parity was not conserved in the weak interactions. Furthermore, Lee and Yang suggested experiments to test this hypothesis. Shortly thereafter, Chien-Shiung Wu performed the experiment (see https://en.wikipedia.org/wiki/Wu_experiment) and confirmed the violation of parity invariance. The following year, Lee and Yang won the Nobel Prize in Physics. A year later, in 1958, Robert Marshak and his student George Sudarshan developed the vector axial-vector theory (V-A) of the weak interactions, which Richard Feynman and Murray Gell-Mann developed further.

We do not know why the weak interactions violate parity conservation, in particular, why they involve both vector ψ¯γμψ and axial-vector ψ¯γμγ5ψ currents, where γ5iγ0γ1γ2γ3. Now that we have a better understanding of the weak interactions, however, it would be simpler to say that the weak interactions treat left and right-handed fields differently. But, the V-A jargon has stuck.

The Glashow-Salam-Weinberg (GSW) Langrangian (really a Lagrangian density) is built using the fields below for the 1st generation of leptons and quarks:

L=(νLeL),eR,Q=(uLdL),uR,dR,where PL=12(1γ5),PR=12(1+γ5)are projection operators that split afermion field into its left and right componentsψL,R=PL,Rψ,

which implies ψ=ψL+ψR. Note the absence, in this model, of the field νR.

The GSW model (before addition of the Higgs field) is defined by the Langrangian,

L=L¯iγμ(μ1+ig1YL2Bμ1+ig2τ2Wμ)L+Q¯iγμ(μ1+ig1YLQ2Bμ1+ig2τ2Wμ)Q+e¯Riγμ(μ+ig1YR2Bμ)eR+u¯Riγμ(μ+ig1YRu2Bμ)uR+d¯Riγμ(μ+ig1YRd2Bμ)dR14WμνWμν14BμνBμν,

where

1 is a 2×2 unit matrix,τ are the Pauli matrices,the Y are called hypercharges,while the tensors Wμν and Bμν are given byWμν=μWννWμg2Wμ×Wν,=μWννWμg2ϵijkWμjWνk,Bμν=μBννBμ and Wμ=(Wμ1,Wμ2,Wμ3).

The quantities enclosed in the first and second pair of parentheses in the Lagrangian are 2×2 matrices that act on the lepton and quark doublet fields L and Q, respectively, while the parentheses in the remaining fermionic terms enclose 1×1 matrices, that is, scalars, which act on the singlet fields eR, uR, and dR, respectively. However, each field is intrinsically a 4-component Lorentz object: a 4-component spinor in the case of the fermionic fields and a 4-vector in the case of the bosonic fields.

In the GSW Lagrangian, the hypercharges are assigned specific numerical values. Therefore, the Lagrangian has only two free parameters, namely, the couplings g1 and g2.

SU(2)U(1) Invariance

The GSW Lagrangian is invariant under the group SU(2)U(1). The elements U of SU(2) can be represented as the 2×2 matrices

U=exp(i(τ/2)θ(x)), whileU=exp(i(Y/2)α(x)),

represent the elements of U(1). The SU(2) elements are defined by three spacetime-dependent functions, θ(x)=(θ1(x), θ2(x), θ3(x)), each associated with group generators T1=τ1/2, T2=τ2/2, and T3=τ3/2, respectively, while the U(1) elements are defined by a single function α(x) and a single generator Y/2.

Invariance under the group SU(2)U(1) means that the following replacements leave the Lagrangian unchanged,

SU(2)LUL,QUQ,eReR,uRuR,dRdR12τWμU(12τWμ)U11ig2(μU)U1,BμBμ,U(1)LUL,QUQeRUeR,uRUuR,dRUdR,WμWμ,BμUBμU11ig1(μU)U1.

It is usually argued that the above local gauge invariance means that the phases of the fields can be set to different values at different events in spacetime and that this makes sense because there is no reason why, for example, Andromedans, ten million years from now, should adopt the same phase convention as Earthlings do now. Personally, I find this argument unconvincing because there are, after all, global symmetries that span the whole of spacetime. For example, the replacements (see for example, https://arxiv.org/pdf/hep-ph/0410370.pdf, p. 9)

Qeiθ/3Q,uReiθ/3uR,dReiθ/3dR,andLeiϕL,eReiϕeR,

which correspond to baryon number and lepton number conservation, respectively, are global symmetries that leave the Lagrangian invariant. These global symmetries were not imposed, but appear for free and, in that sense, are accidental. My question is: why is it fine to have global symmetries where certain quantities, such as θ and ϕ must be the same everywhere and everywhen, that is, Andromedans and Earthlings must use the same values of the phases even though they have never met, and yet it is not fine to have the same gauge field phases everywhere? There is no good reason that I can discern. I think it is better simply to accept the discovery that local guage invariance yields theories, which for reasons not yet clear, are spectacularly successful.

Hypercharges

The hypercharge Y, electric charge Q (in units of the proton charge and not to be confused with the quark doublet), and the third component of the isospin I3=±1/2 are related by the Gell-Mann Nishijima relation Y=2(QI3). For the left-handed neutrino field, Q=0 and I3=+1/2, therefore, the hypercharge is YL=1. For the left-handed electron field, Q=1 and I3=1/2 and again YL=1. On the other hand, since the right-handed electron field is an SU(2) singlet and, therefore, transforms trivially, that is, eReR, I3=0 and therefore YR=2. We conclude that the hypercharge is a property of the doublet or singlet.

In the quark model, up quarks have electric charge Q=+2/3 (in units of the proton charge), while down quarks have electric charge Q=1/3. Therefore, in order to satisfy the Gell-Mann Nishijima relation we must make the assignments YLQ=1/3, YRu=4/3, and YRd=2/3.

Bosons

If we define Xμ=(τ/2)Wμ and Xμν=(τ/2)Wμν, and noting that τaτb=iϵabcτc+δab1 and, therefore, Tr(τaτb)=2δab, we can write

Xμν=μXννXμ+ig2[Xμ,Xν],
and, therefore,
WμνWμν=12Tr(XμνXμν).

In the low energy limit (q2<<MW, where the W boson mass MW80GeV), the GSW model reproduces the Fermi model for β-decay provided that we set

GF2=g228MW2.

In [1]:
import sympy as sm
import sympy.functions.special.tensor_functions as t
from sympy.matrices import Matrix
from sympy.physics.matrices import mgamma, msigma
from sympy import I, trigsimp, radsimp

# enable pretty printing of equations
sm.init_printing()

levi = t.LeviCivita
kron = t.KroneckerDelta

Define fields

Alas, there are differing conventions for defining the fields. In the textbook by Mark Thomson, Wμ±=12(Wμ1Wμ2), while in the textbook by Gordon Kane (Modern Elementary Particle Physics, 2nd Edition) these fields are defined by Wμ±=12(Wμ1±Wμ2). And in the book by Donnelly, Formaggio, Holstein, Milner, and Surrow (Foundations of Nuclear and Particle Physics), the convention is Wμ±=12(Wμ1±Wμ2). Here, we'll follow the convention in Thomson's book.

In [2]:
gamma  = sm.Symbol('\gamma^{\mu}', commutative=False)
nubarL = sm.Symbol('\overline{\\nu}_L', commutative=False)
nuL    = sm.Symbol('\\nu_L', commutative=False)
ebarL  = sm.Symbol('\overline{e}_L', commutative=False)
eL     = sm.Symbol('e_L', commutative=False)
ebarR  = sm.Symbol('\overline{e}_R', commutative=False)
eR     = sm.Symbol('e_R', commutative=False)

ubarL  = sm.Symbol('\overline{u}_L', commutative=False)
uL     = sm.Symbol('u_L', commutative=False)
ubarR  = sm.Symbol('\overline{u}_R', commutative=False)
uR     = sm.Symbol('u_R', commutative=False)

dbarL  = sm.Symbol('\overline{d}_L', commutative=False)
dL     = sm.Symbol('d_L', commutative=False)
dbarR  = sm.Symbol('\overline{d}_R', commutative=False)
dR     = sm.Symbol('d_R', commutative=False)

sbarL  = sm.Symbol('\overline{s}_L', commutative=False)
sL     = sm.Symbol('s_L', commutative=False)
sbarR  = sm.Symbol('\overline{s}_R', commutative=False)
sR     = sm.Symbol('s_R', commutative=False)

cbarL  = sm.Symbol('\overline{c}_L', commutative=False)
cL     = sm.Symbol('c_L', commutative=False)
cbarR  = sm.Symbol('\overline{c}_R', commutative=False)
cR     = sm.Symbol('c_R', commutative=False)

B, W1, W2, W3         = sm.symbols("B_\mu, W^1_\mu, W^2_\mu, W^3_\mu")
Bnu, W1nu, W2nu, W3nu = sm.symbols("B_\\nu, W^1_\\nu, W^2_\\nu, W^3_\\nu")
Wplus, Wminus         = sm.symbols("W^+_\mu, W^-_\mu")
Wplusnu,Wminusnu,W3nu = sm.symbols("W^+_\\nu, W^-_\\nu, W^3_\\nu")

Z, A     = sm.symbols('Z_\mu, A_\mu')
Znu, Anu = sm.symbols('Z_\\nu, A_\\nu')

nuL, eL, eR, uL, uR, dL, dR, B, W1, W2, W3
Out[2]:
(νL, eL, eR, uL, uR, dL, dR, Bμ, Wμ1, Wμ2, Wμ3)
In [3]:
Wplus, Wminus, Z, A
Out[3]:
(Wμ+, Wμ, Zμ, Aμ)

Define couplings, hypercharges, and weak mixing angle θW

In [4]:
g1, g2, thetaW     = sm.symbols('g_1 g_2 \\theta_W')
YQL, YdL, YuR, YdR = sm.symbols('Y^Q_L Y^d_L Y^u_R Y^d_R')
YL, YR, YphiL      = sm.symbols('Y_L Y_R Y_L^\phi')
g1, g2, YL, YR, YQL, YuR, YdR, YphiL, thetaW
Out[4]:
(g1, g2, YL, YR, YLQ, YRu, YRd, YLϕ, θW)

Define SU(2) lepton and quark doublets L and Q

In [5]:
L = Matrix([nuL, eL])
Lbar = Matrix([nubarL, ebarL]).T

Q = Matrix([uL, dL])
Qbar = Matrix([ubarL, dbarL]).T

Lbar, L, Qbar, Q
Out[5]:
([ν¯Le¯L], [νLeL], [u¯Ld¯L], [uLdL])

Define τW matrix

In [6]:
tau1 = msigma(1)
tau2 = msigma(2)
tau3 = msigma(3)
w    = tau1*W1 + tau2*W2 + tau3*W3
w
Out[6]:
[Wμ3Wμ1iWμ2Wμ1+iWμ2Wμ3]

The fields Wμ1Wμ2 have the form of charged fields, which we write as (Wμ1Wμ2)/2Wμ±, and are identified as the fields of the charged weak currents.

In [7]:
w = w.subs({W1 - I*W2: sm.sqrt(2)*Wplus,
            W1 + I*W2: sm.sqrt(2)*Wminus})
w
Out[7]:
[Wμ32Wμ+2WμWμ3]

Define matrix b

In [8]:
b = B*tau1**2 
b
Out[8]:
[Bμ00Bμ]

Compute matrices

ML=g1YL2Bμ+g2τ2Wμ

and

MQ=g1YLQ2Bμ+g2τ2Wμ
In [9]:
ML = (g1/2)*YL*b + (g2/2)*w

MQ = (g1/2)*YQL*b + (g2/2)*w

ML, MQ
Out[9]:
([BμYLg12+Wμ3g222Wμ+g222Wμg22BμYLg12Wμ3g22], [BμYLQg12+Wμ3g222Wμ+g222Wμg22BμYLQg12Wμ3g22])

Expand

L¯γμMLLQ¯γμMQQ

Note: we had to switch the order of gamma and Lbar for the algebra to look correct.

In [10]:
a = sm.expand(-gamma*Lbar*ML*L)[0] + sm.expand(-gamma*Qbar*MQ*Q)[0]
a
Out[10]:
BμYLQg1d¯LγμdL2BμYLQg1u¯LγμuL2BμYLg1ν¯LγμνL2BμYLg1e¯LγμeL22Wμ+g2ν¯LγμeL22Wμ+g2u¯LγμdL22Wμg2d¯LγμuL22Wμg2e¯LγμνL2Wμ3g2ν¯LγμνL2+Wμ3g2d¯LγμdL2+Wμ3g2e¯LγμeL2Wμ3g2u¯LγμuL2

Add the singlet terms

$- \frac{g_1}{2} YR B\mu \bar{e}_R \gamma^\mu e_R

  • \frac{g_1}{2} Y^uR B\mu \bar{u}_R \gamma^\mu u_R
  • \frac{g_1}{2} Y^dR B\mu \bar{d}_R \gamma^\mu d_R$
In [11]:
l = a
l = l - (g1/2)*YR*B*ebarR*gamma*eR 
l = l - (g1/2)*YuR*ubarR*gamma*B*uR
l = l - (g1/2)*YdR*dbarR*gamma*B*dR
l
Out[11]:
BμYLQg1d¯LγμdL2BμYLQg1u¯LγμuL2BμYRdg1d¯RγμdR2BμYRug1u¯RγμuR2BμYLg1ν¯LγμνL2BμYLg1e¯LγμeL2BμYRg1e¯RγμeR22Wμ+g2ν¯LγμeL22Wμ+g2u¯LγμdL22Wμg2d¯LγμuL22Wμg2e¯LγμνL2Wμ3g2ν¯LγμνL2+Wμ3g2d¯LγμdL2+Wμ3g2e¯LγμeL2Wμ3g2u¯LγμuL2

The unification of the electromagnetic and weak nuclear forces

The key idea in the GSW model, as in the original 1961 model by Glashow, is the unification of the electromagnetic and weak interations, which is expressed as follows

(BμWμ3)=(cosθWsinθWsinθWcosθW)(AμZμ).

What the unification means concretely is that above a certain critical temperature, the symmetry between the two forces (called the electroweak symmetry) is restored and the two forces merge into a single electroweak force. Below the critical temperature, the electroweak symmetry is said to be spontaneously broken by the change in shape of the potential energy of the Higgs field. While the ensemble of infinitely many vacuum state solutions of the theory respects the electroweak symmetry, the particular vacuum state into which the universe settles does not.

We should be thankful for broken symmetry because without it the universe would be far less diverse and we would not exist.

In [12]:
AZv = Matrix([A, Z])
BWv = Matrix([B, W3])
rot = Matrix([[sm.cos(thetaW),  sm.sin(thetaW)],
             [-sm.sin(thetaW),  sm.cos(thetaW)]])
irot= rot**-1
irot.simplify()
BW  = irot * AZv
AZ  = rot  * BWv
AZ, BW
Out[12]:
([Bμcos(θW)+Wμ3sin(θW)Bμsin(θW)+Wμ3cos(θW)], [Aμcos(θW)Zμsin(θW)Aμsin(θW)+Zμcos(θW)])

Collect terms for Aμ and Zμ

In [13]:
l = l.subs(B,  BW[0])
l = l.subs(W3, BW[1])
l = sm.expand(l)
l = sm.collect(l, A)
l = sm.collect(l, Z)
l
Out[13]:
Aμ(YLQg1cos(θW)d¯LγμdL2YLQg1cos(θW)u¯LγμuL2YRdg1cos(θW)d¯RγμdR2YRug1cos(θW)u¯RγμuR2YLg1cos(θW)ν¯LγμνL2YLg1cos(θW)e¯LγμeL2YRg1cos(θW)e¯RγμeR2g2sin(θW)ν¯LγμνL2+g2sin(θW)d¯LγμdL2+g2sin(θW)e¯LγμeL2g2sin(θW)u¯LγμuL2)2Wμ+g2ν¯LγμeL22Wμ+g2u¯LγμdL22Wμg2d¯LγμuL22Wμg2e¯LγμνL2+Zμ(YLQg1sin(θW)d¯LγμdL2+YLQg1sin(θW)u¯LγμuL2+YRdg1sin(θW)d¯RγμdR2+YRug1sin(θW)u¯RγμuR2+YLg1sin(θW)ν¯LγμνL2+YLg1sin(θW)e¯LγμeL2+YRg1sin(θW)e¯RγμeR2g2cos(θW)ν¯LγμνL2+g2cos(θW)d¯LγμdL2+g2cos(θW)e¯LγμeL2g2cos(θW)u¯LγμuL2)

In the above expression there is an interaction between the electromagnetic field Aμ and the neutrino current ν¯LγμνL. However, we have no evidence that such an interaction exists. Therefore, we need to get rid of it. This can be done by setting YLg1cosθW = g2sinθW.

In [14]:
l = l.subs(-YL*g1*sm.cos(thetaW), g2*sm.sin(thetaW))
l
Out[14]:
Aμ(YLQg1cos(θW)d¯LγμdL2YLQg1cos(θW)u¯LγμuL2YRdg1cos(θW)d¯RγμdR2YRug1cos(θW)u¯RγμuR2YRg1cos(θW)e¯RγμeR2+g2sin(θW)d¯LγμdL2+g2sin(θW)e¯LγμeLg2sin(θW)u¯LγμuL2)2Wμ+g2ν¯LγμeL22Wμ+g2u¯LγμdL22Wμg2d¯LγμuL22Wμg2e¯LγμνL2+Zμ(YLQg1sin(θW)d¯LγμdL2+YLQg1sin(θW)u¯LγμuL2+YRdg1sin(θW)d¯RγμdR2+YRug1sin(θW)u¯RγμuR2+YLg1sin(θW)ν¯LγμνL2+YLg1sin(θW)e¯LγμeL2+YRg1sin(θW)e¯RγμeR2g2cos(θW)ν¯LγμνL2+g2cos(θW)d¯LγμdL2+g2cos(θW)e¯LγμeL2g2cos(θW)u¯LγμuL2)

So far, so good. Now we wish to recover the known form of the electromagnetic interaction, which for electrons is qAμ(e¯LγμeL+e¯RγμeR). We can do this if we impose the conditions

(YR/2)g1cosθW=g2sinθW=q,

where q is the electric charge, or equivalently,

cosθW=qg1(YR/2) and sinθW=qg2.

The above expressions imply

q=g1g2(YR/2)g12(YR/2)2+g22,tanθW=(YR/2)g1g2
In [15]:
l = l.subs({-YR*g1*sm.cos(thetaW): 2*g2*sm.sin(thetaW) })
l
Out[15]:
Aμ(YLQg1cos(θW)d¯LγμdL2YLQg1cos(θW)u¯LγμuL2YRdg1cos(θW)d¯RγμdR2YRug1cos(θW)u¯RγμuR2+g2sin(θW)d¯LγμdL2+g2sin(θW)e¯LγμeL+g2sin(θW)e¯RγμeRg2sin(θW)u¯LγμuL2)2Wμ+g2ν¯LγμeL22Wμ+g2u¯LγμdL22Wμg2d¯LγμuL22Wμg2e¯LγμνL2+Zμ(YLQg1sin(θW)d¯LγμdL2+YLQg1sin(θW)u¯LγμuL2+YRdg1sin(θW)d¯RγμdR2+YRug1sin(θW)u¯RγμuR2+YLg1sin(θW)ν¯LγμνL2+YLg1sin(θW)e¯LγμeL2+YRg1sin(θW)e¯RγμeR2g2cos(θW)ν¯LγμνL2+g2cos(θW)d¯LγμdL2+g2cos(θW)e¯LγμeL2g2cos(θW)u¯LγμuL2)

Simplify expressions

We extract the coefficients of the bosonic fields, that is, the currents, and manipulate them into the forms in which they are typically expressed. One of the key manipulations is substituting the appropriate expressions for the hypercharges.

Hypercharge assignments

Assume the validity of the relation Y=2(QI3), where Q and I3, respectively, are the electric charge in units of the proton charge and the 3rd component of the isospin. Setting Q=0 for the upper component of the lepton doublet, L, which is occupied by the neutrino field, and noting that I3=+1/2 for that component, we find YL=1. Repeating this for the lower component, Q=1 and I3=1/2, we again obtain YL=1. And similarly for YR with Q=1 and I3=0, we find YR=2. Also, YLQ=2(QfI3f)=1/3, YRu=2Qu=4/3, where f is the fermion flavor, and YRd=2Qd=2/3.

In [16]:
q = sm.symbols('q')
Qf, Qu, Qd, Qe, Qnu = sm.symbols('Q^f, Q^u, Q^d, Q^e, Q^\\nu')
If3, Id3, Iu3, Ie3, Inu3 = sm.symbols('I^f_3, I^d_3, I^u_3, I^e_3, I^\\nu_3')
q, Qf, Qu, Qd, Qe, Qnu, If3, Id3, Iu3, Ie3, Inu3
Out[16]:
(q, Qf, Qu, Qd, Qe, Qν, I3f, I3d, I3u, I3e, I3ν)

Get coefficient of Wμ+

In [17]:
Wpluscoeff  = l.coeff(Wplus)
Wpluscoeff  = Wpluscoeff.subs(g2, q/sm.sin(thetaW)).factor(q).simplify()
Wpluscoeff
Out[17]:
2q(ν¯LγμeL+u¯LγμdL)2sin(θW)

Get coefficient of Wμ

In [18]:
Wminuscoeff = l.coeff(Wminus)
Wminuscoeff = Wminuscoeff.subs(g2, q/sm.sin(thetaW)).factor(q).simplify()
Wminuscoeff
Out[18]:
2q(d¯LγμuL+e¯LγμνL)2sin(θW)

Get coefficient of Aμ

In [19]:
N, E, U, D = sm.symbols('N, E, U, D')

Acoeff = l.coeff(A).expand()

Acoeff = Acoeff.subs({ebarL*gamma*eL: E,
                      ubarL*gamma*uL: U, 
                      dbarL*gamma*dL: D})
Acoeff = Acoeff.collect(E).collect(U).collect(D)

Acoeff = Acoeff.subs({E:ebarL*gamma*eL, 
                      U: ubarL*gamma*uL, 
                      D: dbarL*gamma*dL})

Acoeff = Acoeff.subs(g1, g2*sm.sin(thetaW)/sm.cos(thetaW))

Acoeff = Acoeff.subs({g2*sm.sin(thetaW): q})

Acoeff
Out[19]:
YRdqd¯RγμdR2YRuqu¯RγμuR2+qe¯LγμeL+qe¯RγμeR+(YLQq2q2)u¯LγμuL+(YLQq2+q2)d¯LγμdL

Implement hypercharge assignments for electromagnetic current.

In [20]:
Acoeff = Acoeff.subs(YuR, 2*Qu)
Acoeff = Acoeff.subs(YdR, 2*Qd)
Acoeff = Acoeff.subs(YQL, 2*(Qf - If3))
Acoeff
Out[20]:
Qdqd¯RγμdRQuqu¯RγμuR+qe¯LγμeL+qe¯RγμeR+(q(2I3f+2Qf)2q2)u¯LγμuL+(q(2I3f+2Qf)2+q2)d¯LγμdL
In [21]:
# up quark current
old = Acoeff.coeff(ubarL*gamma*uL)
new = old.subs({2*If3: 1, Qf: Qu})
Acoeff = Acoeff.subs({old: new})

# down quark current
old = Acoeff.coeff(dbarL*gamma*dL)
new = old.subs({2*If3:-1, Qf: Qd})
Acoeff = Acoeff.subs({old: new}).simplify()

Acoeff
Out[21]:
q(Qdd¯LγμdLQdd¯RγμdRQuu¯LγμuLQuu¯RγμuR+e¯LγμeL+e¯RγμeR)

From the above that the electromagnetic current Jemμ can be written as

Jemμ=qf=u,d,eQfψ¯fγμψf,
where ψ=ψL+ψR, noting that PL,RγμPL,R=0.

Get coefficient of Zμ

Try some algebraic manipulations to simplify expression. Unfortunately, collect does not work on non-commutative symbols. So try the following hack.

In [22]:
Zcoeff = l.coeff(Z)

Zcoeff = Zcoeff.subs({nubarL*gamma*nuL: N, 
                      ebarL*gamma*eL: E,
                      ubarL*gamma*uL: U, 
                      dbarL*gamma*dL: D})
Zcoeff = Zcoeff.collect(E).collect(N).collect(U).collect(D)

Zcoeff = Zcoeff.subs({N:nubarL*gamma*nuL, 
                      E:ebarL*gamma*eL, 
                      U: ubarL*gamma*uL, 
                      D: dbarL*gamma*dL})
Zcoeff = Zcoeff.subs(g1, g2*sm.sin(thetaW)/sm.cos(thetaW))

Zcoeff
Out[22]:
YRdg2sin2(θW)d¯RγμdR2cos(θW)+YRug2sin2(θW)u¯RγμuR2cos(θW)+YRg2sin2(θW)e¯RγμeR2cos(θW)+(YLQg2sin2(θW)2cos(θW)g2cos(θW)2)u¯LγμuL+(YLQg2sin2(θW)2cos(θW)+g2cos(θW)2)d¯LγμdL+(YLg2sin2(θW)2cos(θW)g2cos(θW)2)ν¯LγμνL+(YLg2sin2(θW)2cos(θW)+g2cos(θW)2)e¯LγμeL

Notice that the coefficients of g2cos(θW) is I3. Therefore, let's make those substitutions in the above.

In [23]:
old = Zcoeff.coeff(ubarL*gamma*uL)
new = old.subs({g2*sm.cos(thetaW)/2: g2*Iu3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(dbarL*gamma*dL)
new = old.subs({g2*sm.cos(thetaW)/2: -g2*Id3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(nubarL*gamma*nuL)
new = old.subs({g2*sm.cos(thetaW)/2: g2*Inu3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarL*gamma*eL)
new = old.subs({g2*sm.cos(thetaW)/2: -g2*Ie3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

Zcoeff
Out[23]:
YRdg2sin2(θW)d¯RγμdR2cos(θW)+YRug2sin2(θW)u¯RγμuR2cos(θW)+YRg2sin2(θW)e¯RγμeR2cos(θW)+g2(2I3νcos2(θW)+YLsin2(θW))ν¯LγμνL2cos(θW)+g2(2I3dcos2(θW)+YLQsin2(θW))d¯LγμdL2cos(θW)+g2(2I3ecos2(θW)+YLsin2(θW))e¯LγμeL2cos(θW)+g2(2I3ucos2(θW)+YLQsin2(θW))u¯LγμuL2cos(θW)

Implement hypercharge assignments for weak neutral current. Note the I3 value for singlet fields is always zero.

Tip: To disable a cell use esc r; use esc y to reactivate it and esc m to go to markdown mode.

In [24]:
old = Zcoeff.coeff(dbarR*gamma*dR)
new = old.subs({YdR: 2*Qd}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ubarR*gamma*uR)
new = old.subs({YuR: 2*Qu}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarR*gamma*eR)
new = old.subs({YR: 2*Qe}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(nubarL*gamma*nuL)
new = old.subs({YL: 2*(Qnu - Inu3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(dbarL*gamma*dL)
new = old.subs({YQL: 2*(Qd - Id3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarL*gamma*eL)
new = old.subs({YL: 2*(Qe - Ie3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ubarL*gamma*uL)
new = old.subs({YQL: 2*(Qu - Iu3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

Zcoeff
Out[24]:
Qdg2sin2(θW)d¯RγμdRcos(θW)+Qeg2sin2(θW)e¯RγμeRcos(θW)+Qug2sin2(θW)u¯RγμuRcos(θW)g2(I3νQνsin2(θW))ν¯LγμνLcos(θW)g2(I3dQdsin2(θW))d¯LγμdLcos(θW)g2(I3eQesin2(θW))e¯LγμeLcos(θW)g2(I3uQusin2(θW))u¯LγμuLcos(θW)

The weak neutral current Jncμ can be written as

Jncμ=2qsin(2θW)(f=d,e,uQfsin2θWψ¯RfγμψRf+f=ν,d,e,u(I3f+Qfsin2θW)ψ¯LfγμψLf).

Final form of GSW lagrangian for 1st generation fermions

Electromagnetic interactions

In [25]:
Acoeff*A
Out[25]:
Aμq(Qdd¯LγμdLQdd¯RγμdRQuu¯LγμuLQuu¯RγμuR+e¯LγμeL+e¯RγμeR)

Weak charged current interactions

In [26]:
Wpluscoeff*Wplus + Wminuscoeff*Wminus
Out[26]:
2Wμ+q(ν¯LγμeL+u¯LγμdL)2sin(θW)2Wμq(d¯LγμuL+e¯LγμνL)2sin(θW)

Weak neutral current interactions

In [27]:
Zcoeff = Zcoeff.factor(g2).subs({g2: q / sm.sin(thetaW)})
Zcoeff*Z
Out[27]:
Zμq(Qdsin2(θW)d¯RγμdR+Qesin2(θW)e¯RγμeR+Qusin2(θW)u¯RγμuR+(I3ν+Qνsin2(θW))ν¯LγμνL+(I3d+Qdsin2(θW))d¯LγμdL+(I3e+Qesin2(θW))e¯LγμeL+(I3u+Qusin2(θW))u¯LγμuL)sin(θW)cos(θW)

The GSW model incorporates quantum electrodynamics (QED), predicts weak charged and neutral current interactions and incorporates the observation that neither the electromagnetic nor the weak neutral currrent interactions change flavor, while the weak charged current interactions always do. And, of course, the model incorporates the parity violation of the weak interactions.

The Higgs Sector

The Higgs Lagrangian is given by

LHiggs=(Dμϕ)(Dμϕ)V(ϕϕ),
where
Dμ=μ+ig1YLϕ2Bμ+ig2τ2Wμ,V(ϕϕ)=μ2ϕϕ+λ(ϕϕ)2,and the 4-component Higgs field is given byϕ=(ϕ1+iϕ2ϕ3+iϕ4).

Qualitative description of the Higgs mechanism

The stability of the system requires λ>0. In the very early universe, we presume that μ2>0, in which case 0|ϕ|0=0, that is, the value of the Higgs field in the vacuum is zero. But, at some critical temperature, Tcrit, there is a phase transition in which μ2 changes sign thereby causing the Higgs potential to develop infinitely many degenerate vacua in each of which the Higgs field is non-zero. Above Tcrit, there is a unique vacuum state and this state respects the symmetry of the Lagrangian. Below Tcrit, the ensemble of vacua respects the symmetry, but, the universe cools into one of the infinitely many vacua, which on its own breaks the symmetry. This is an example of spontaneous symmetry breaking, here electroweak symmetry breaking (EWSB). Spontaneous symmetry breaking is entirely analogous to the breaking of the O(3) symmetry in an (infinite) ferromagnet as it cools below its critical temperature and develops a magnetization in one of infinitely many possible directions, that is, "vacuum" states of the ferromagnet.

The vacuum state of the universe is one in which the Higgs field behaves like a superconductor that attenuates the propagation of the Wμ and Zμ fields, thereby rendering these fields massive, but permits the free propagation of the Aμ. (The theory is made to do this!) Moreover, the interactions between the Higgs field and the fermions cause a rapid flipping between the left-handed and right-handed fermion fields, which builds up energy within them that we interpret as their mass. The results from the LHC experiment are, so far, consistent with this picture.

Since there is no right-handed neutrino the flipping mechanism cannot occur and therefore the neutrino remains massless in the GSW model. However, neutrinos are now known to have (very small) masses, but, how their masses arise remains an open question.

Quantum numbers

We need to specify the value of the hypercharge YLϕ associated with the doublet ϕ. As above, the hypercharge is chosen so that the Gell-Mann Nishijima relation Y=2(QI3) holds, where the third component of the weak isospin I3=+1/2 for the upper component of a doublet and I3=1/2 for the lower component. In the GSW model, it is assumed that the vacuum expectation of ϕ is

0|ϕ|0=12(0v).
Since the vacuum has a net electric charge of zero, we must set Q=0 for the lower component ϕ3+iϕ4. From YLϕ=2(QI3) it follows that YLϕ=+1. This also shows that the upper component, ϕ1+iϕ2 must be assigned Q=+1, which of course is why it is assumed to be zero in the vacuum otherwise the entire universe would have a huge net positive charge!

Higgs field at low energies

At energies not too far from the vacuum state, we assume that the Higgs field can be approximated as follows

ϕH=12(0v+h),
where h represents the fluctuations of the Higgs field about its average value v246GeV in the vacuum state (the vacuum expectation value). The quanta of the field h are identified with the Higgs bosons discovered at CERN in 2012.

In the following, we shall also need the conjugate field defined by

Hciτ2H=12(v+h0).
In [28]:
v = sm.symbols('v')
h = sm.symbols('h')
H = Matrix([0, (v+h)/sm.sqrt(2)])
F = (g1/2)*YphiL*b + (g2/2)*w
Hc=  sm.I*tau2*H
F, H, Hc
Out[28]:
([BμYLϕg12+Wμ3g222Wμ+g222Wμg22BμYLϕg12Wμ3g22], [02(h+v)2], [2(h+v)20])

Note

The matrix F is Hermitian, that is, F=F.

In [29]:
y    = F*H
ybar = H.T*F
f = ybar*y
f = f[0]
f = f.subs(YphiL, 1)
f
Out[29]:
Wμ+Wμg22(h+v)24+(h+v)2(Bμg12Wμ3g22)22

Vector boson - Higgs boson interactions

In [30]:
f = f.subs(g1*B/2 - g2*W3/2, Z * sm.sqrt(g1**2 + g2**2)/2)
f
Out[30]:
Wμ+Wμg22(h+v)24+Zμ2(g12+g22)(h+v)28

The terms involving the Higgs boson field are the single and di-Higgs interactions, while the term v2 is the mass term. For a massive vector boson, Vμ, the mass term has the form (mV2/2)VμVμ. The term Wμ+Wμ is the sum of two terms quadratic in the fields W1 and W2, with the same mass, so the mass term has the form mW2Wμ+Wμ. Therefore, in the GSW model the W boson the mass is predicted to be mW=vg2/2, while for the Z boson the prediction is mZ=vg12+g22/2 and, therefore, mW/mZ=g2/g12+g22=cosθW, which modulo higher order corrections agrees with measurements as shown below.

In [31]:
sw2 = 0.2397
sw = sm.sqrt(sw2)
cw = sm.sqrt(1-sw2)
print('cos(theta_W) = %8.3f' % cw)
cos(theta_W) =    0.872
In [32]:
MW=80.385
MZ=91.188
cwp = MW/MZ
print('M_W / M_Z    = %8.3f\t%8.3f' % (cwp, cw / cwp))
M_W / M_Z    =    0.882	   0.989

The Higgs potential

V(H)=μ2HHλ(HH)2

In [33]:
mu, lm = sm.symbols('\mu, \lambda')
V = mu*mu*H.T*H - lm*(H.T*H)**2
V = V.expand()[0]
V
Out[33]:
λh44λh3v3λh2v22λhv3λv44+μ2h22+μ2hv+μ2v22

h-h interactions at low energies

In [34]:
V.subs(mu**2, v**2*lm)
Out[34]:
λh44λh3vλh2v2+λv44

Yukawa interactions between fermions and the Higgs field

2ge(L¯HeR+e¯RHL)2gu(Q¯HcuR+u¯RHcQ)2gd(Q¯HdR+d¯RHQ)

Consider the dimensions of the above Yukawa interactions. In natural units, the Lagrangian has units of [M]4. Since a fermion mass term looks like

mψ¯ψ

and dim(m)=[M], it follows that a fermion field has dimensions [M]3/2. Since the Higgs boson is a scalar field, it has dimensions [M]. Consequently, the Yukawa interactions between fermions and the Higgs boson must be a gauge invariant combination of two fermion fields and the Higgs field. Also, because the Higgs field is a doublet, it is necessary to have it interact with the Dirac conjugate of a doublet field and a singlet field.

Notice also that in order to avoid a flavor changing Yukawa interaction, we need to fudge the interaction of the up quark field with the Higgs field by using the conjugate field Hc rather than H!

In [35]:
ge, me = sm.symbols('g_e, m_e')
gu, mu, gd, md = sm.symbols('g_u, m_u, g_d, m_d')
gu, mu, gd, md
ge, me, gu, mu, gd, md
Out[35]:
(ge, me, gu, mu, gd, md)
In [36]:
Hc = sm.I*tau2*H
Qbar, Q, Hc, Hc.T
Out[36]:
([u¯Ld¯L], [uLdL], [2(h+v)20], [2(h+v)20])
In [37]:
Le = -ge*(Lbar*H*eR + ebarR*H.T*L)[0]*sm.sqrt(2)
Le = Le.expand()
Le = Le.simplify().expand().collect(h*ge).collect(ge*v)
Le = Le.subs(ge*v, me)
Le = Le.subs(ge, me/v)

Lq = -gd*(Qbar*H*dR  + dbarR*H.T*Q)[0]*sm.sqrt(2) \
     -gu*(Qbar*Hc*uR + ubarR*Hc.T*Q)[0]*sm.sqrt(2)
Lq = Lq.expand()
Lq = Lq.subs(gu*v, mu)
Lq = Lq.subs(gd*v, md)
Lq = Lq.collect(gd*h)
Lq = Lq.collect(gu*h)
Lq = Lq.collect(md)
Lq = Lq.collect(mu)
Lq = Lq.subs(gu, mu/v)
Lq = Lq.subs(gd, md/v)
Lh = Le + Lq
Lh
Out[37]:
hmd(d¯RdLdRd¯L)v+hme(e¯ReLeRe¯L)v+hmu(u¯RuLuRu¯L)v+md(d¯RdLdRd¯L)+me(e¯ReLeRe¯L)+mu(u¯RuLuRu¯L)

We see that the Yukawa couplings, together with the particular choice of the value of the Higgs field in the vacuum state, yields two important consequences as alluded to above: mass terms for the fermions and interactions between the Higgs boson and fermions that are proportional to the fermion mass and inversely proportional to the vacuum expectation value v. We conclude that the Higgs mechanism generates mass terms for the weak vector bosons as well as the fermions while preserving guage invariance.

The Cabibbo Hypothesis

So far we have considered only the 1st generation of particles and we have implicitly assumed that there is no "crosstalk" between the generations. However, there are decays that can be explained if "crosstalk" exists between generations. For example, the existence of decays such as

Λ(d,u,s)p(d,u,u)eν¯e
are readily explained by supposing that the W boson can convert s-quarks to u-quarks. Further clues are provided by the rates of decays such as
π+μ+νμ,K+μ+νμ,
which are measured to be in the ratio
Γ(π+μ+νμ)Γ(K+μ+νμ)120.
The above decays can be described with the following Feynman diagrams, decays in which the couplings g2? differ from g2 by the appropriate factor in order to match the measured ratio. These observations and many others were explained by the Italian physicist Cabibbo who suggested that the lower component of the quark doublet, that is, the down quark field, be replaced with the combination
dLdL=cosθCdL+sinθCsL,
where θC is called the Cabibbo angle. With a suitable choice of the Cabibbo angle (about 12o), the Cabibbo hypothesis worked well. However, as is evident from the form of the down component of the left-handed part of weak neutral current,
(I3dQdsin2(θW))d¯LγμdL,
the Cabibbo hypothesis predicts the existence of flavor-changing neutral currents (FCNC) at a rate in sharp disagreement with observations.

In [38]:
thetaC = sm.symbols('theta_C')
xbarL, xL = sm.symbols('\overline{d^\prime}_L, '\
                       'd^\prime_L', commutative=False)
Nc = xbarL * gamma * xL 
Nc = Nc.subs({xL: sm.cos(thetaC)*dL + sm.sin(thetaC)*sL,
              xbarL: sm.cos(thetaC)*dbarL + sm.sin(thetaC)*sbarL})
Nc = Nc.expand()
Nc = (Id3 - Qd * sm.sin(thetaW)**2) * Nc
Nc
Out[38]:
(I3dQdsin2(θW))(sin2(θC)s¯LγμsL+sin(θC)cos(θC)d¯LγμsL+sin(θC)cos(θC)s¯LγμdL+cos2(θC)d¯LγμdL)

The GIM Mechanism

In 1970, Sheldon Glashow, John Iliopoulos and Luciano Maiani (GIM) noted that if the strange quark were the lower component of another doublet,

(cLsL),
whose upper component would be a new quark field, dubbed "charm", and if one generalized Cabibbo's idea and assumed that the lower components of the two quark doublets were mixed as follows
(dLsL)=(cosθCsinθCsinθCcosθC)(dLsL),
then the replacement
sLsL=sinθCdL+cosθCsL
would yield exact cancellation of the unwanted FCNC interactions, as demonstrated below.

Implement GIM Mechanism

In [39]:
ybarL, yL = sm.symbols('\overline{s^\prime}_L, '\
                                  's^\prime_L', 
                                  commutative=False)
Nc += (Id3 - Qd * sm.sin(thetaW)**2) * ybarL * gamma * yL
Nc
Out[39]:
(I3dQdsin2(θW))s¯LγμsL+(I3dQdsin2(θW))(sin2(θC)s¯LγμsL+sin(θC)cos(θC)d¯LγμsL+sin(θC)cos(θC)s¯LγμdL+cos2(θC)d¯LγμdL)
In [40]:
Nc = Nc.subs({yL:-sm.sin(thetaC)*dL + sm.cos(thetaC)*sL,
              ybarL:-sm.sin(thetaC)*dbarL + sm.cos(thetaC)*sbarL})
Nc.simplify()
Out[40]:
(I3dQdsin2(θW))(d¯LγμdL+s¯LγμsL)

By introducing a 2nd doublet, the GIM mechanism neatly renders the weak neutral currents diagonal again and, moreover, predicts the existence of a 4th quark, the charm quark. This prediction was spectacularly confirmed in 1974 with the discovery of the J/ψ at Brookhaven and SLAC.

Unfortunately, however, no one understands why mixing across the (three) generations should exist and what determines the strength of the mixing.

Vector boson self-interactions

The pure vector boson Lagrangian is given by

LVB=14WμνWμν14BμνBμν,
where
(Wμν)i=(μWννWμ)ig2ϵijkWμjWνk,Bμν=μBννBμ.

From

(BμWμ3)=(cosθWsinθWsinθWcosθW)(AμZμ)Wμ±=12(Wμ1Wμ2)
we obtain
W1=(W+W+)/2W2=(WW+)/(2)W3=sinθWA+cosθWZB=cosθWAsinθWZ,
which we use below to rewrite the vector boson Lagrangian in terms of the usual fields.

In [41]:
Bmu, W1mu, W2mu, W3mu = sm.symbols("B_\mu, W^1_\mu, W^2_\mu, W^3_\mu",
                                   commutative=False)
Bnu, W1nu, W2nu, W3nu = sm.symbols("B_\\nu, W^1_\\nu, W^2_\\nu, W^3_\\nu",
                                   commutative=False)
Wplusmu, Wminusmu     = sm.symbols("W^+_\mu, W^-_\mu",
                                   commutative=False)
Wplusnu, Wminusnu     = sm.symbols("W^+_\\nu, W^-_\\nu",
                                   commutative=False)
dmu, dnu              = sm.symbols('\partial_\mu, \partial_\\nu', 
                                   commutative=False)
Zmu, Amu              = sm.symbols('Z_\mu, A_\mu',
                                   commutative=False)
Znu, Anu              = sm.symbols('Z_\\nu, A_\\nu',
                                   commutative=False)
dmuW1nu = dmu*W1nu
dmuW2nu = dmu*W2nu
dmuW3nu = dmu*W3nu

dnuW1mu = dnu*W1mu
dnuW2mu = dnu*W2mu
dnuW3mu = dnu*W3mu

dmuBnu  = dmu*Bnu
dnuBmu  = dnu*Bmu
In [42]:
Vmu = Matrix([W1mu, W2mu, W3mu])
Vnu = Matrix([W1nu, W2nu, W3nu])

dVmu= Matrix([dmuW1nu, dmuW2nu, dmuW3nu])
dVnu= Matrix([dnuW1mu, dnuW2mu, dnuW3mu])
WW  = dVmu - dVnu - g2*Vmu.cross(Vnu)
WW  = WW.subs({W1mu: (Wminusmu + Wplusmu)/sm.sqrt(2), 
               W1nu: (Wminusnu + Wplusnu)/sm.sqrt(2),
               W2mu: (Wminusmu - Wplusmu)/sm.sqrt(2), 
               W2nu: (Wminusnu - Wplusnu)/sm.sqrt(2),
               W3mu: sm.sin(thetaW)*Amu + sm.cos(thetaW)*Zmu,
               W3nu: sm.sin(thetaW)*Anu + sm.cos(thetaW)*Znu}).expand()

BB  = dmuBnu - dnuBmu
BB  = BB.subs({Bmu: sm.cos(thetaW)*Amu - sm.sin(thetaW)*Zmu,
               Bnu: sm.cos(thetaW)*Anu - sm.sin(thetaW)*Znu}).expand()
VV  = BB*BB +  (WW.T*WW)[0]
VV  = VV.expand()
VV = trigsimp(VV.subs({g2: q / sm.sin(thetaW)}))
VV
Out[42]:
q2AμWν+AμWν+q2AμWν+Wμ+Aν+q2AμWνAμWνq2AμWνWμAνq2Wμ+AνAμWν++q2Wμ+AνWμ+Aνq2WμAνAμWν+q2WμAνWμAνq2AμWν+Wμ+Zνtan(θW)+q2Aμ(Wν+)2Zμtan(θW)q2AμWνWμZνtan(θW)+q2Aμ(Wν)2Zμtan(θW)q2Wμ+AνZμWν+tan(θW)q2Wμ+ZνAμWν+tan(θW)+q2(Wμ+)2AνZνtan(θW)+q2(Wμ+)2ZνAνtan(θW)q2WμAνZμWνtan(θW)q2WμZνAμWνtan(θW)+q2(Wμ)2AνZνtan(θW)+q2(Wμ)2ZνAνtan(θW)q2ZμWν+Wμ+Aνtan(θW)+q2Zμ(Wν+)2Aμtan(θW)q2ZμWνWμAνtan(θW)+q2Zμ(Wν)2Aμtan(θW)q2Wμ+ZνZμWν+tan2(θW)+q2(Wμ+)2Zν2tan2(θW)q2WμZνZμWνtan2(θW)+q2(Wμ)2Zν2tan2(θW)q2ZμWν+Wμ+Zνtan2(θW)q2ZμWνWμZνtan2(θW)+q2Zμ2(Wν+)2tan2(θW)+q2Zμ2(Wν)2tan2(θW)+q2Wμ+WνWμ+Wνsin2(θW)q2Wμ+WνWμWν+sin2(θW)q2WμWν+Wμ+Wνsin2(θW)+q2WμWν+WμWν+sin2(θW)qAμWν+μWν+qAμWν+νWμ+qAμWνμWν+qAμWννWμ++qWμ+AνμWνqWμ+AννWμqWμ+WνμAν+qWμ+WννAμqWμAνμWν++qWμAννWμ++qWμWν+μAνqWμWν+νAμqμAνWμ+Wν+qμAνWμWν++qμWν+AμWνqμWν+WμAνqμWνAμWν++qμWνWμ+Aν+qνAμWμ+WνqνAμWμWν+qνWμ+AμWν+qνWμ+WμAν+qνWμAμWν+qνWμWμ+AνqWμ+WνμZνtan(θW)+qWμ+WννZμtan(θW)+qWμ+ZνμWνtan(θW)qWμ+ZννWμtan(θW)+qWμWν+μZνtan(θW)qWμWν+νZμtan(θW)qWμZνμWν+tan(θW)+qWμZννWμ+tan(θW)qZμWν+μWνtan(θW)+qZμWν+νWμtan(θW)+qZμWνμWν+tan(θW)qZμWννWμ+tan(θW)qμWν+WμZνtan(θW)+qμWν+ZμWνtan(θW)+qμWνWμ+Zνtan(θW)qμWνZμWν+tan(θW)qμZνWμ+Wνtan(θW)+qμZνWμWν+tan(θW)+qνWμ+WμZνtan(θW)qνWμ+ZμWνtan(θW)qνWμWμ+Zνtan(θW)+qνWμZμWν+tan(θW)+qνZμWμ+Wνtan(θW)qνZμWμWν+tan(θW)+μAνμAνμAννAμ+μWν+μWν+μWν+νWμ++μWνμWνμWννWμ+μZνμZνμZννZμνAμμAν+νAμνAμνWμ+μWν++νWμ+νWμ+νWμμWν+νWμνWμνZμμZν+νZμνZμ

This is a complicated expression, but some conclusions are readily apparent such as the prediction of 3 and 4 vector boson interactions.

The QCD Lagrangian

The Lagrangian of quantum chromodynamics (QCD), for the 1st generation of particles, is given by

LQCD=U¯iγμ(μ1+ig3λ2Gμ)U+D¯iγμ(μ1+ig3λ2Gμ)D

where U and D, respectively, are the up and down quark field triplets each of whose components is associated with one of three quark colors, λa are the 8 Gell-Mann matrices each associated with one of the 8 gluon fields Gμa, and g3 is the strong coupling parameter. The quantities in parentheses are 3×3 matrices. The QCD Lagrangian is invariant under SU(3) transformations.

When QCD is added to the GSW (electroweak) Lagrangian, thereby forming the Lagrangian of the Standard Model (SM), the up and down quark fields acquire a 3-color index. Therefore, in the electroweak part of the Lagrangian each quark field will now carry a color index, in which case it is necessary to sum over the color indices in that part of the Lagrangian. Note that each quark field in the QCD Lagrangian is the sum of left-handed and right-handed fields.

Since the SM is gauge invariant, it is necessary to fix the definition of its bosonic (gauge) fields by choosing a gauge. Choosing a gauge is akin to choosing a coordinate system in that the predictions of the theory do not depend on the choice of gauge, at least in the limit of exact calculations. (In practice, calculations are done using perturbation theory. Consequently, depending on how the calculations are done there may be some residual, unphysical, gauge dependence in the predictions.) Unfortunately, choosing the gauge requires the addition of quite a bit of extra structure, involving unphysical fields called ghost fields, which makes what is otherwise an elegant structure much less so.

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